Question

The sum of two positive numbers is 12. What is the smallest possible value of the sum of their squares?

Answer 1

Let the two positive numbers be 'a' and 'b' and 'S" be the sum of the squares.

\(a + b = 12 \)
\(S = a^2+b^2\) = \(a^2+(12-a)^2\)
Simplify the right hand side. It represents a parabola. The minimum S value occurs at the lowest point of the parabola, namely, the vertex in this case.
Find the vertex and that will be the 'a' value for which S is minimum.

Answer 2

Refer to the Mathematical attachment.